Concept# Bisection

Summary

In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a 'bisector'. The most often considered types of bisectors are the 'segment bisector' (a line that passes through the midpoint of a given segment) and the 'angle bisector' (a line that passes through the apex of an angle, that divides it into two equal angles).
In three-dimensional space, bisection is usually done by a bisecting plane, also called the 'bisector'.
Perpendicular line segment bisector
Definition
*The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly.
*The perpendicular bisector of a line segment AB also has the property that each of its points X is equidistant from segment AB's endpoints:
(D)\quad |XA| = |XB|.
The proof follows from and Pythagoras' theorem:
:|X

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We define the bisector energy E(P) of a set P in R-2 to be the number of quadruples (a, b, c, d) is an element of P-4 such that a, b determine the same perpendicular bisector as c, d. Equivalently, E(P) is the number of isosceles trapezoids determined by P. We prove that for any epsilon > 0, if an n-point set P has no M(n) points on a line or circle, then we have E(P) = O(M(n)(2/5) n(12/5+epsilon) + M(n)n(2)). We derive the lower bound E(P) = Omega(M(n)n(2)), matching our upper bound when M(n) is large. We use our upper bound on E(P) to obtain two rather different results: (i) If P determines O(n/root log n) distinct distances, then for any 0 < alpha